Class 10 Mathematics • Full Test Paper Set 2

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x) = x^2 - p(x+1) - c$, then the value of $(\alpha+1)(\beta+1)$ is:

The LCM of the smallest two-digit composite number and the smallest composite number is:

If $\sin \theta + \cos \theta = \sqrt{2} \cos \theta$, then the value of $\tan \theta$ is:

In $\triangle ABC$, $DE \parallel BC$ such that $AD=x$, $DB=x-2$, $AE=x+2$ and $EC=x-1$. The value of $x$ is:

The coordinates of the point which is equidistant from the three vertices of the $\triangle AOB$ as shown in the figure (where $A=(2x,0)$ and $B=(0,2y)$) is:

If two positive integers $a$ and $b$ are written as $a = x^3y^2$ and $b = xy^3$, where $x, y$ are prime numbers, then $HCF(a, b)$ is:

The roots of the equation $x^2 + x - (a+1)(a+2) = 0$ are:

If the distance between the points $(4, p)$ and $(1, 0)$ is 5, then the value of $p$ is:

In a circle of diameter 42 cm, if an arc subtends an angle of $60^\circ$ at the centre, then the length of the arc is:

A card is drawn from a well-shuffled deck of 52 cards. The probability that the card will not be an ace is:

If the mean of the following distribution is 2.6, then the value of $y$ is:
Variable (x): 1, 2, 3, 4, 5
Frequency (f): 4, 5, y, 1, 2

The value of $k$ for which the system of equations $x + 2y = 5$ and $3x + ky + 15 = 0$ has no solution is:

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is:

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. The volume of the solid in terms of $\pi$ is:

If $2 \sin 2\theta = \sqrt{3}$, then the value of $\theta$ is:

The probability of getting a bad egg in a lot of 400 is 0.035. The number of bad eggs in the lot is:

If the point $P(k-1, 2)$ is equidistant from the points $A(3, k)$ and $B(k, 5)$, then the values of $k$ are:

In the given figure, if $\angle AOB = 125^\circ$, then $\angle COD$ is equal to:

Assertion (A): The number $5^n$ cannot end with the digit 0 for any natural number $n$.
Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.

Assertion (A): The point $(0, 4)$ lies on the y-axis.
Reason (R): The x-coordinate of a point on the y-axis is zero.

Solve for $x$: $\frac{1}{a+b+x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}$, where $a+b \neq 0$.

In the given figure, $PA$ and $PB$ are tangents to the circle with centre $O$ such that $\angle APB = 50^\circ$. Write the measure of $\angle OAB$.

Find the mode of the following distribution:
Class: 0-20, 20-40, 40-60, 60-80, 80-100
Frequency: 10, 8, 12, 16, 4

If $\tan(A+B) = \sqrt{3}$ and $\tan(A-B) = \frac{1}{\sqrt{3}}$; $0^\circ < A+B \le 90^\circ$; $A > B$, find $A$ and $B$.

Find the ratio in which the y-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$. Also find the point of intersection.

Prove that $\sqrt{5}$ is an irrational number.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, find the value of $k$.

A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \csc \theta$.

Prove that the parallelogram circumscribing a circle is a rhombus.

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

State and prove Basic Proportionality Theorem. Using it, prove that the diagonals of a trapezium divide each other proportionally.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

The median of the following data is 525. Find the values of $x$ and $y$, if the total frequency is 100.
Class Interval: 0-100, 100-200, 200-300, 300-400, 400-500, 500-600, 600-700, 700-800, 800-900, 900-1000
Frequency: 2, 5, x, 12, 17, 20, y, 9, 7, 4

Case Study 1:
India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in the 6th year and 22600 in the 9th year.
1. Find the production in the 1st year.
2. Find the production in the 8th year.
3. Find the total production in first 3 years.

Case Study 2:
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th the distance AD on the eighth line and posts a red flag.
1. What is the distance between both the flags?
2. If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
3. If the race is extended to 200 pots, what will be the coordinate of the red flag?

Case Study 3:
A satellite flying at height $h$ is watching the top of the two tallest mountains in Uttarakhand and Karnataka, them being Nanda Devi (height 7,816m) and Mullayanagiri (height 1,930m). The angles of depression from the satellite, to the top of Nanda Devi and Mullayanagiri are $30^\circ$ and $60^\circ$ respectively. If the distance between the peaks of two mountains is 1937 km, and the satellite is vertically above the midpoint of the distance between the two mountains.
1. Find the distance of the satellite from the top of Nanda Devi.
2. Find the distance of the satellite from the top of Mullayanagiri.
3. Find the height of the satellite from the ground.